An efficient simulated annealing algorithm for a No-Wait Two Stage Flexible Flow Shop Scheduling Problem

International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 4, August 2011
DOI : 10.5121/ijait.2011.1402 13
An efficient simulated annealing algorithm for a
No-Wait Two Stage Flexible Flow Shop
Scheduling Problem
M. Rabiee1
, P. Ramezani2
and R. Shafaei3
1
Department of Industrial Engineering, K. N. Toosi University of Technology,
Tehran, Iran
meysam_rabiee@yahoo.com
2
Tehran, Iran
prh_79@yahoo.com
3
Tehran, Iran
shafaei@kntu.ac.ir
ABSTRACT
In this paper, no wait two stage flexible flow shop scheduling problem (FFSSP) is solved using two meta-
heuristic algorithms. This problem with minimum makespan performance measure is NP-Hard. The
proposed algorithms are Simulated Annealing and Genetic Algorithm. The results are analyzed in terms
of Relative Percentage Deviation of Makespan. The performance of the proposed algorithms are studied
and compared with that of MDA algorithm. For this propose a number of problems in different sizes are
solved. The results of the studies proposes the effective algorithm. This is followed by describing the
outline of the study, concluding remarks and suggesting potential areas for further researches
KEYWORDS
No wait, Two stage, Flexible flow shop, Makespan, Simulated annealing, Genetic algorithm.
1. INTRODUCTION
Flexible flow shop is a typical classical flow shop, which consists of m stages each one with one
or more than one identical parallel machines, n given jobs to be processed on m subsequent
stages. . Flexible flow shop also known as a hybrid flow shop, flexible flow line or flow shop
scheduling problem with parallel machines . This area has been studied by many researchers [1-
4]. For a literature review in this area the readers are referred to those conducted by Ruiz Et al
[5] and Ribas Et al [6].
A prominent class of scheduling problems is identified by a no-wait production environment, in
which there is no storage between the machines. Thus, jobs must be processed from the start to
finish, deprived of any interruption on machines or between them. Consequently, the processing
of a job on the initial machine may need to be delayed to guarantee that no waiting takes place
on any subsequent machine. There are two main motives for a no wait environment: the type of
procedure, or a lack of storage between intermediate machines (work stations). In some
industries, due to the temperature or other characteristics of the material it is required that each
operation follow the previous one immediately. Such situations appear in the chemical
processing [11], food processing[12], concrete ware production [13], pharmaceutical
processing [14] and production of steel, plastics, aluminum products [15].

14
These kinds of manufactures can often be modeled as a ‘pure’ flow shop environment in which
all of n jobs follow the same sequence at each stage, each of n jobs consist of two operations
owning a predetermined processing order through machines. Each job is to be processed
without preemption and interruption on two stages or between them. That is, once a job is
started on the first machine, it has to be continuously processed through the subsequent machine
without interruption. In addition, each machine can handle no more than one job at a time and
each job has to visit each machine exactly once. Therefore, it may occur a condition that the
starting time of a job on the first machine must be delayed in order to meet the no-wait
requirement [16]. As an example, in a chemical industry if the waiting time is allowed between
each subsequent stage, it may lead to deteriorate the material property (e.g. degrading the
polymer). Therefore, in such industries, once an operation is completed, the subsequent
operation shall be started without any delay. For a further study in no wait manufacturing the
readers are recommended to review the paper present by Hall and Sriskandarajah [12]
The problem studied in this paper is makespan minimization in a no wait two stages flexible
flow shop scheduling problem as it has a multiplicity of practical implementations in both
manufacturing industry and service corporations. This particular hybrid flow shop(i.e. two stage
hybrid flow shop) scheduling problem with the objective of minimizing makespan is first
considered in Riane et al. [7]. Narasimhan,S.& Mangiameli,P [8] also introduced a heuristic
algorithm known as the Cumulative Minimum Deviation (CMD) rule for a two-stage hybrid
flow shop scheduling problem with single machine in stage one and two independent machines
in stage 2. The results of the study reveals that this algorithm outperforms SPT and LPT in
reducing the machine idle time and in-process waiting time at the second stage.
Liu Zhixin et al [17] presented a heuristic algorithm named Least Deviation (LD) rule for two-
stage no-wait hybrid flow shop scheduling with a single machine in each stage and makespan
performance measure. The results of the study indicate that this algorithms is superior than L,D
and Johnson. Also they shown LD algorithm has low computational complexity and is easy to
implementation thus it is of favourable application value. Jinxing Xie et al. [18] proposed a new
heuristic algorithm known as Minimum Deviation Algorithm (MDA) to minimize makespan in
a two stage flexible flow shop with no waiting time.Experimental results of their study show
that MDA outperformed partition method, partition method with LPT, Johnson’s as well as the
modified Johnson’s algorithms.
Jinxing Xie and Xijun Wang [19] considers the two-stage flexible flow shop scheduling
problem with availability constraints. They discussed the complexity and the approximability of
the problem and provide some approximation algorithms with worst case performance bounds
for some special cases of the problem. Their results showed that the problems studied are much
more difficult to approximate than the case without availability constraints. Huang et al [20]
considered a no-wait two stage flexible flow shop with setup times and with minimum total
completion time performance measure. They proposed an integer programming model and ant
colony optimization heuristic approach. The ACO results revealed that the efficiency of the
proposed algorithm is superior to those solved by integer programming while having
satisfactory solutions.
In this study a no wait flexible flow shop problem is solved using two methaheuristic algorithms
based on Simulated Annealing and Genetic Algorithm. The aim is to find an effective algorithm
with minimum makespan. The rest of the article is formed as follows: Section 2 briefly
describes the problem addressed in this paper. Section 3 presents the framework of the proposed
algorithms. Numerical problems developed to study the performance of the proposed algorithms
are introduced in section 4. This is followed by presenting the results of simulation study.
Finally Section 5 describes the outline of the research with the concluding remarks and
suggestion for further researches.

15
2. PROBLEM DEFINITION
The no-wait flexible flow shop scheduling problem (NWFFSSP) can be described as follows:
given the processing time p(i, j) of Stage i on Job j, each of n jobs will be sequentially processed
in stage 1, 2 respectively. At each stage there are mi machines. In addition at any event of time,
each machine can process at most one job. Similarly each operation of a job can only be
processed on one machine. Once the sequence of the jobs at the first stage is defined, the same
sequence is applied for the second stage. To fulfill the no-wait restrictions, the completion time
of a job on a given machine must be equal to the start time of the job on the next machine. In
other words, there must be no waiting time between the processing of any consecutive
operations of each job. The problem is to find a sequence that the maximum completion time,
i.e., makespan (Cmax), is minimized.
Let 1 2{ , ,..., }nπ π π π= denote the schedule or permutation of jobs to be processed, 1( , )j jL π π−
the minimum delay on the first machine between the start of job 1jπ − and jπ restricted by the
no-wait constraint. Then L can be calculated as follows:
1
1 1 1
2
2 1
( , ) ( ,1) max[0,max{ ( ,1) ( , )}]. (1)
k k
j j j j j
k s
h h
L P p p hπ π π π π
−
− − −
≤ ≤
= =
= + −∑ ∑
Thus, the makespan can be defined as max 1
2
( ) ( , ) ( ) (2)
n
j j sum n
j
C L Pπ π π π−
=
= +∑
Where
1
( ) ( , ).
m
sum n n
k
P p kπ π
=
= ∑
The no-wait FFSSP with the makespan criterion is to find a permutation *
π among the all
permutations Π such that *
maxarg{ ( )} min .Cπ π π= → ∀ ∈Π Where arg represents the
arranged permutation.
The problem is shown by 2 1 2 max( , ) |no-wait| CF m m . If the number of parallel machines in each
stage is not input variables, the problem can be shown by 2 max( ) |no-wait| CF p . In this study the
operations' set up times are assumed to be independent from the job sequences and hence is
added to the operation times. This problem is schematically depicted in Figure 1.
Figure 1. Schematic of the problem
As illustrated above, each job has 1 2m m× possible schedules and hence n jobs have !n possible
schedules. Therefore in total there are 1 2! n n
n m m× × possible solutions for this problem [21].
The two stage no wait flexible flow shop problems are NP-hard in the strong sense [22].
Therefore, all exact algorithms for even a simple flow shop and simple parallel machines will

16
most likely have running times that increase exponentially with the problem Size. In this paper
we propose metaheuristic algorithms to the problem described above. The framework of this
algorithm is explained in the next section.
3. FRAMEWORK OF THE PROPOSED ALGORITHMS
The proposed algorithms in this study are based on GA and SA methods. They are categorized
based on the algorithms applied for job sequencing and machine assignment. With reference to
the algorithms presented in this paper, the job sequences are generated using the stochastic
search process of SA and GA. The jobs are then allocated to the machines using a constructive
heuristic algorithm which allocates the job with the first priority to the machine with the earliest
available time.
In the remaining part of this section, first the schematic of a chromosome/ solution is presented.
Then the structure of the Genetic Algorithm and Simulated Annealing are presented. This is
followed by describing the constructive heuristic algorithm is applied for machine assignment in
the metaheuristic algorithms.
3.1. The representation of chromosome/solution
3.1.1. GA and SA chromosome/solution representation
Suppose that there are n jobs each one to be assigned to two identical machines in each stage.
The proposed algorithms first choose the initial solutions using a uniform distribution in a range
between 0 and one. For the sake of clarification, suppose that there are 5 jobs and two machines
in each stage. At first 5 random numbers are generated in a range between 0 and 1. Assume that
the outcomes are 0.45, 0.63, 0.13, 0.33 and 0.77. Figure 2 shows the corresponding
chromosome/solution.
Figure.2 Initialized chromosome for GA and SA
The job sequences are defined according to the ascending values of the chromosome/solutions
shown in Figure 3. Figure 6 shows the resulting sequence.
Figure 3. The Job sequence corresponding the GA/SA algorithm
Having generated the job sequences, at each stage the jobs are assigned to the machines using
the constructive heuristic algorithm accordingly. It is described at below.
0.45 0.63 0.13 0.33 0.77
3 4 1 2 5

17
3.2. Constructive heuristic
The jobs are assigned to the machine with the earliest available time. The details are explained
at below.
1 1 2 2
2 2 1 1 2 1
1 2
0 : 0 ( 1, 2, ..., ) an d 0 ( = 1 ,2 ,..., )
m in ( ) , m in ( ) ,
{ , , ..., } as jo b seq u en ce an d Q to set o f sch ed u led jo b s
S tep 1 : select th e m ach in e w ith earli
i l
l i
n
S tep S et t i m t l m
t T t T T T T
S et π π π π
= = =
= = = −
=
2
1 1 1 1 2 1 2
1
est availab le tim e fo r b o th stag es
an d co m p u te acco rd in g ly. th e jo b seq u en ce.
{ } an d ad d to th e
S tep 2 : if 0 th en an d
if 0 th e
j
J j j j j j j
J
t
S et j j Q
P T t t p t t p
P T
π π→ −
− ≥ = + = +
− < 1 2 2 2 2n an d
S tep 3 : if , sto p an d co m p u te o b jective fu n ctio n . o th erw ise , g o
to step 1
j j j j jt t t t p
π
= = +
= ∅
Where 1 2andi lt t are current processing times in first stage and second stage respectively,
1 2andj jP P represent processing time of the job j in first stage and second stage respectively.
And π denotes the job permutation.
3.3. Genetic Algorithm
Holland [23] for first time introduced genetic algorithm as an optimization tool that redefines
the problems in the genotype space by converting the features of phenotype as the
chromosomes. In the genetic algorithm a population of chromosomes is maintained and
stochastic search operators such as crossover, mutation and selection are used to finding better
answers [24]. The combinations of genes are evolved through the genetic operators so that the
chromosomes move toward the optimal solution generation by generation. There are three
genetic operators: selection, crossover and mutation. Crossover operators can produce new
solutions optimistically retaining good characteristics from parents, and mutation operator can
enhance diversity and provide a chance to escape from local optima. So far, GAs have been
widely applied in many fields, especially in operations research [25, 26].
3.3.1. Selection operator
The main goal of selection operators is choosing better chromosomes to obtain better solutions.
To achieve this goal, several issues should be taken into consideration; how to choose
chromosomes in the population in order to create offspring and how many offspring should be
created and how chromosomes are selected as population in the next generation. [27]
Parent selection is performed by roulette wheel selection scheme. In this procedure selection
probability of each chromosome is calculated by following formula:
max
1
fitness value
C
= (1-3)
In this scheme each chromosome has more chance to be selected as the fitness value increases.
This rule results in finding strong off springs.

18
After determination of chromosome as parents, in order to transit to the next generation (λ µ+
)-selection strategy is applied as survivor selection rule [28]. In the process of survival selection,
in each generation, after producing λ offspring and calculating the respective fitness values,
among λ µ+ chromosomes, the best λ µ+ chromosomes are selected as the next population
members.
3.3.2. Crossover
The crossover operation is used to effectively explore the search space. The main purpose of
crossover operator is to exchange information among randomly selected parent chromosomes
with the aim of producing better solution, i.e. offspring. It recombines genetic members of two
parent chromosomes to produce the offspring for the next generation that retain good properties
of the parent chromosomes. The exchange is also intended to search for better genes [29]. To
perform a crossover, two parents are selected from the mating pool at random. With the
probability of 1 − pc, the parents are copied as they are. On the other hand, with the probability
of pc, called the crossover probability, the crossover operation is performed. Figure 4 presents
the crossover operation.
Figure 4. Crossover procedure
According to the above addressed figure, firstly an array of binary mask with a length equal to
the chromosome members is generated. Then the offspring is generated from a couple of
nominated parents in two ways resulting in offspring 1 and 2. In offspring one, the ith member
is copied from the ith member of parent 1 if the respective binary mask member value is equal
to one, otherwise the value is copied from parent 2. This is continued till all members of the
offspring are constructed. The second offspring member generation is similar to that of the first
offspring except that, in any selection, the member from the first parent is selected if the
respective binary mask value is equal to zero, otherwise the member from the second value is
generated. Syswerda, G [30] shown that this crossover operator obtains good exploitations.
Therefore in this paper it is applied as a cross over operator in the proposed GA algorithm.
3.3.3. Mutation
Mutation serves to maintain diversity in the population and is a way of enlarging the
exploration. It acts to prevent the selection and crossover from focusing on a narrow area of the
search space or the GA getting stuck in a local optimum. Mutation is done by selecting two
different locations on the chromosome at random and interchanging the jobs at these locations.
For each child obtained from crossover, the mutation operator is applied independently with a
small probability pm. A sample of swap mutation is shown in the Figure 5.
Binary mask 1 0 1 0 0 1 0 1
Parent 1 0.51 0.34 0.09 0.76 0.42 0.23 0.81 0.15
Parent 2 0.43 0.86 0.52 0.29 0.16 0.91 0.73 0.31
Offspring 1 0.51 0.86 0.09 0.29 0.16 0.23 0.73 0.15
Offspring 2 0.43 0.34 0.52 0.76 0.42 0.91 0.81 0.31

19
Figure 5. Mutation procedure
3.3.4. Termination criterion
The algorithm continues until all generations are created. The running time of the GA depends
on the number of generations along with the population size. The GA will have a greater like
hood that an optimal or near to optimal solution is found if it is to run longer. Structure of
genetic algorithm which is used in this study are shown in Figure 6.
Figure 6. Proposed genetic algorithm structure
3.4. Simulated annealing algorithm
The Simulated Annealing (SA) is a search technique, which can be used to seek good solutions
for various combinatorial problems. It has its origin in the material science and physics. The
Parent 0.51 0.34 0.09 0.76 0.42 0.23 0.81 0.15
Offspring 0.51 0.34 0.23 0.76 0.42 0.09 0.81 0.15
No
Yes
Return the best found solution
Machine assignment using constructive heuristic
Generate initial sequences
Fitness assignment of initial population
Is gen < genmax?
Crossover
Mutation
Fitness evaluation
Form the new population
gen=gen+1

20
motivation for simulated annealing comes from an analogy between the physical annealing of
solid materials and optimization problem.
The analogy between physical annealing and simulated annealing can be summarized as
follows: (i) The physical configurations or states of the molecules correspond to optimization
solution; (ii) the energy of molecules corresponds to the objective function or cost function; (iii)
a low energy sate corresponds to an optimal solution; (iv) the cooling rate corresponds to the
control parameter which will affect the acceptance probability. The algorithm consists of four
main components: (i) configurations; (ii) re-configuration technique; (iii) cost function and (iv)
cooling schedule [31, 32]. Similar to genetic algorithm, simulated annealing is a stochastic
search method. It aims to find an acceptable solution where it is impractical to find the optimum
solution using other methods.
Simulated Annealing is a generalization of a Monte Carlo method for examining the equations
of state and frozen states of a system [32]. Since the original work conducted by Metropolis on
simulated annealing [32], a significant amount of works have been done on this technique and
its applications in different fields. Because of the combinatorial nature of the scheduling
problems, application of simulated annealing on this problem has been investigated extensively.
Examples of using SA can be found in single machine, Flow shop, Job shop problems and
combinations of these problems with various restrictions and job conditions [33-35]. However,
to the author’s knowledge, so far no work was conducted to solve no wait two stage flexible
flow shop scheduling problem using simulated annealing approach.
The detailed description of the simulated annealing algorithm to solve no wait two stage flexible
flow shop problems with minimum makespans is presented below.
• Step 1: Set the parameters: In this step the initial parameters of the algorithm are set.
These include initial temperature, T (1), the minimum temperature, T (min), the
temperature reduction multiplier, α, and the number of iterations at each Temperature,
N. Also the iteration counter, n, are set to one in order to start the first iteration.
• Step 2: Set F(X) equal to the objective function for the initial solution: In this step the
value of the objective function for the initial sequence (X) obtained in the first phase is
set equal to F(X). X is also defined as the best solution found (Xbest) and the best
objective function calculated, F(Xbest), is set equal to F(X).
• Step 3: Generate Y, a neighbourhood solution: In this step two jobs are randomly
chosen and swapped in the sequence of jobs that define their priorities. Following the
procedure used in phase 1, find the corresponding solution. Denote this solution as Y
and the value of its objective function as F(Y).
• steps 4: Check for improvement: The improvement, δ , of the new solution compared to
the previous solution is evaluated as the difference between the objective functions of
solutions Y and X. If δ is less than zero, then there is an improvement and the algorithm
continues with step 5 otherwise go to step 6.
• Steps 5: Compare the new solution with the best solution found. If the new solution is
better, i.e. F(Y) <F (Xbest), replace the best solution and its objective function with Y
and F(Y) respectively. Continue by following step 7.
• Steps 6: Accept or reject the non-improving move randomly In order to randomly
accept a non-improving move that might lead to a better solution, calculate L=Exp (-
δ/T(t)) and compare it to R, a random number between zero and one. If L>R the non-
improving move will be accepted with a hope to find a better neighbor of the solution,
and the algorithm continues with step 7. Otherwise, solution Y is ignored and the
algorithm returns to step 3 to generate a new solution.
• Step 7: Update solution X Since solution Y was accepted in steps 5 or 6, solution X and
F(X) are replaced with Y and F(Y) respectively.
• Step 8: Update and check the iteration counter and compare the counter n with N. If
n<N, increase n by one and go to step 3 to start a new iteration, otherwise increase the

21
temperature counter t by one, calculate the new temperature: ( )T t T(t 1)α= × −
, set
n=1, and continue with step 9.
• Step 9: Check the termination criteria; If the temperature of the system is less than or
equal to the minimum temperature allowed, the algorithm is terminated. Otherwise
continue with the new temperature iteration by going to step 3. Structure of proposed
simulated annealing is shown in Figure 7.
3.5. The fitness function
The performance measure considered in this study is minimum makespan. The corresponding
fitness function is calculated as follow:
Cj= Completion time of job j
Makespan= Cmax = max (Cj)
Figure 7. Proposed simulated annealing structure
Yes
Yes
No
Is it accepted?
Fitness evaluation
Set the solution as the best
Is the max
generation is
met?
Lower temperature
Reach to the
T ?
Produce new
solution by
neighborhoo
d mechanism
Return the best found solution
Generate initial sequence
Machine assignment using constructive
No
No
Yes

22
4. NUMERICAL EXPERIMENTS
In this section, the performances of the proposed algorithms are studied by solving series of
problems. At first the parameters used in the simulation experiments are introduced. This
includes the scale of the problems in terms of the number of the jobs, number of machines at
each stage and processing times. Next the results of the simulation study are illustrated and the
performance of the proposed algorithms is compared with that of the others by solving various
problems in different sizes.
4.1. Parameters of the simulation model
In this study we investigated the performance of the proposed approach for a fairly large
number of the problems. In terms of the number of machines, the problems studied in this
research are classified into two main categories namely small and large problem. Table.1 shows
the problem size for each category.
Table.1 Problem size
Problem Number of machines
Small 1 2 1 2 1 2( 3, 4),( 2, 2),( 3, 2)m m m m m m= = = = = =
Large 1 2 1 2 1 2( 8, 10),( 10, 10),( 12, 10)m m m m m m= = = = = =
For each problem illustrated in Table 2 (i.e. 1 23, 4m m= =
), three different size of orders (i.e.
number of jobs) are considered. They are twice, 4 times and 6 times of total number of machine
respectively. In addition the operation times are generated using a uniform distribution with a
couple of ranges. Table.2 shows the number of jobs and operations distribution.
Table.2 Number of jobs and distribution of processing times
N(number of
jobs)
Small&Large 1 2 1 2 1 2(2 ( )),(4 ( )),(6 ( ))m m m m m m× + × + × +
DT(distribution
of processing
times)
Small (4,40)U
Large (60,300)U
4.2. Parameter tuning
Regarding the metaheuristic algorithms, the value of the parameters used in each algorithm
affect its performance. For this purpose, the simulation was performed for a wide range of the
parameters' values for both GA and SA algorithms and the values corresponding to the best
solutions are selected. Table.3 shows the summary of the tuned parameters' value for the
proposed algorithms.

23
Table. 3 Parameters setting for the algorithms
Parameters Small size Large size
GA Population size 100 200
Generation number 200 500
Crossover rate 0.7 0.7
Swapping mutation rate 0.4 0.4
SA 0T 200 400
fT 0.001 0.0001
Iteration number 200 500
α 0.9 0.95
4.3. The simulation process
This section describes the results of the computational experiments performed to evaluate the
performance of the proposed algorithms compared with that of MDA which is considered as an
efficient algorithm for a no wait two stage flexible flow shop. The model was developed using
MATLAB 2008a in a personal computer with 3.4 GHz, 895 MB memory. The performance of
the algorithms was testified by solving 36 different problems (18 problems in small scale and 18
problems in large scale). Regarding the performance measure, Relative Percentage Deviation
(RPD) of Makespan over the best solution is used. It is calculated as follows:
100sol sol
sol
Method Best
RPD
Best
−
= ×
(3)
Where solMethod is value of method and solBest is the best value between the algorithms.
4.4. Simulation results
As described in section one, review of the literature reveals that MDA obtains good solutions
for the problems studied in this paper. Therefore in our study the performance of the proposed
algorithms were compared with each other and MDA algorithm for 36 different problems.
Simulation results are shown in two scales separately. (See Table.4 and Table.5)
Results shown in Table 4 indicate that for small problems, in 17 out of 18 cases, the Simulated
Annealing (SA) outperforms the other the algorithms. In addition RPD performance measure
with % 95 confidence interval was calculated in both small and large scale problems. Table.6
shows the results. This indicates that in terms of RPD and with %95 confidence interval the
range of the results obtained using the algorithms are not in overlapped. In addition the
performance of the
SA algorithm has lower Confidence interval. This confirms the superiority of the proposed SA
compared with the other algorithms.
Table 5 presents the results for large scale problems. Similar to the small scale problems, here
SA also outperforms the all other algorithms; also similar to small scale problems in terms of
RPD and with %95 confidence interval the range of the results obtained using the algorithms are
not in overlapped. So this confirms the superiority of the proposed SA compared with the other
algorithms

24
Table 4 and 5 also present the performance of each algorithm in terms of CPU time. The results
indicate that MDA has normally yielded the shortest CPU time. Such a processing speed is
obviously due to the fact that MDA is a one iteration simple algorithm compared with the other
algorithms in which the solutions are obtained though multi iteration based rules. Despite the
fact that the proposed SA has significantly longer CPU time than that of MDA, but comparable
with that of the other algorithms, its CPU time still is short. For instance for the largest problem
with 132 jobs and 12 and 10 machines in stage one and two respectively, its CPU time is 230
second which is fairly short. Furthermore, for statistical evaluating the algorithms, mean and
interval ( at 95% confidence interval) are plotted for small and large scale problems in Figure 8
and 9,respectively. with respect to these Figures it could be seen poroposed simulated annealing
outperform GA and MDA in both scales. So, As a result, the proposed SA can be considered as
an effective algorithm for no wait two stage flexible flow shop scheduling problem with
minimum makespan.
Table.4 simulation results for small scale
Cmax Cpu time(s)
no.
jobs M1 M2 SA GA MDA SA GA MDA
8
3 4 78 78 100 3.031 3.469 0.049
2 2 116 116 121 3.031 3.453 0.046
3 2 115 115 126 2.031 3.453 0.048
10
3 4 104 104 128 2.234 1.938 0.054
2 2 150 153 165 2.344 1.938 0.051
3 2 144 143 154 2.266 1.938 0.053
14
3 4 108 109 145 2.859 1.953 0.061
2 2 157 160 188 2.813 1.984 0.057
3 2 133 133 137 2.813 1.969 0.058
16
3 4 117 118 171 2.828 2.578 0.063
2 2 175 179 218 2.938 1.984 0.063
3 2 156 156 170 2.906 2.578 0.063
20
3 4 174 177 214 3.719 3.875 0.066
2 2 261 269 300 3.703 3.283 0.064
3 2 234 236 247 3.781 4.359 0.066
24
3 4 194 196 216 4.250 5.031 0.072
2 2 289 304 305 4.344 5.047 0.069
3 2 254 256 273 4.438 5.031 0.071
Figure 8. Means and interval plot for small scale problems in terms of RPD

25
Table 5. Simulation results for large scale problems
Cmax
Cpu
time(s)
no. jobs M1 M2 SA GA MDA SA GA MDA
72
8 10 1822 1856 2106 41.016 38.438 0.091
10 10 1500 1602 1782 34.734 38.641 0.103
12 10 1395 1468 1571 35.719 41.313 0.164
80
8 10 1959 2014 2261 38.000 42.406 0.168
10 10 1634 1724 1901 37.641 43.516 0.254
12 10 1546 1609 1719 37.766 43.937 0.350
88
8 10 2214 2268 2543 81.344 71.672 0.447
10 10 1840 1950 2153 82.922 73.750 0.485
12 10 1691 1799 1854 82.938 72.797 0.511
108
8 10 2454 2550 2655 141.172 100.438 0.577
10 10 2074 2231 2303 149.922 123.453 0.635
12 10 1966 2075 2128 147.672 293.156 0.646
120
8 10 2784 2949 3005 330.792 106.938 0.666
10 10 2450 2609 2582 169.109 106.750 0.746
12 10 2321 2439 2470 166.828 108.438 0.772
132
8 10 3153 3282 3496 229.859 196.625 0.867
10 10 2656 2850 3031 225.563 197.219 0.945
12 10 2493 2611 2715 230.547 210.844 1.020
Figure 9. Means and interval plot for large scale problems in terms of RPD
4. CONCLUSION
This paper studied the performance of the proposed four metaheuristic algorithms for a no wait
flexible flow shop scheduling problem with minimum makespan performance measure. The
performances of these algorithms were also compared with that of MDA. A simulation model
was developed to study the performance of the algorithms. For this purpose, 36 problems in

26
different sizes in terms of the number of job and number of machine in each stage were solved.
The results of the simulation study revealed that the proposed Simulated Annealing Algorithms
(SA) outperforms the other algorithms in terms of minimum makespan. Such superiority
becomes more significant as the problem size increases. Therefore the proposed SA can be
considered as an efficient algorithm for a no wait two stage flexible flow shop. As a further
research it is recommended to study the performance of the proposed algorithm for the problem
with more than two stage. In addition it is worthwhile to investigate the performance of the
proposed algorithms for the problems with sequence depended set up times.
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Authors:
Meysam Rabiee completed his B.Sc. in Industrial Eng. at Bu-Ali Sina
University (2005_2009), and M.Sc. in Industrial Eng. at K.N. Toosi University
Technology (2009_2011). His research interests are scheduling, intelligent
systems, metaheuristic and heuristic algorithms, prediction and vehicle routing
problems.

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Pezhman Ramezani completed his B.Sc. in Industrial Eng. At Kerman, Iran
(2003-2007) and M.Sc. in Industrial Eng. at K.N. Toosi University Technology,
Tehran,Iran (2009_2011). His research interests are production scheduling,
continuous optimization, financial engineering, multi-criteria decision making
and supply chain management.
Rasoul Shafaei completed his B.Sc. in Isfahan university of technology (in iran)
and his M.Sc in Tarbiat moddaress university (in iran) and his Phd in UMIST
(in united kingdom, Manchester). Currently, he is an Asistant Professor at
Industrial Eng. Department, K.N.Toosi University of technology, Iran. His
research interests are supply chain management, ERP, scheduling and Strategy
Management.

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